Power series rings and projectivity

We show that a formal power series ring $A[[X]]$ over a noetherian ring $A$ is not a projective module unless $A$ is artinian. However, if $(A,{\mathfrak m})$ is local, then $A[[X]]$ behaves like a projective module in the sense that $Ext^p_A(A[[X]], M)=0$ for all ${\mathfrak m}$-adically complete $A$-modules. The latter result is shown more generally for any flat $A$-module $B$ instead of $A[[X]]$. We apply the results to the (analytic) Hochschild cohomology over complete noetherian rings.Comment: Mainly thanks to remarks and pointers by L.L.Avramov and S.Iyengar, we added further context and references. To appear in Manuscripta Mathematica. 7 page

New free divisors from old

We present several methods to construct or identify families of free divisors such as those annihilated by many Euler vector fields, including binomial free divisors, or divisors with triangular discriminant matrix. We show how to create families of quasihomogeneous free divisors through the chain rule or by extending them into the tangent bundle. We also discuss whether general divisors can be extended to free ones by adding components and show that adding a normal crossing divisor to a smooth one will not succeed

Factoring the Adjoint and Maximal Cohen--Macaulay Modules over the Generic Determinant

A question of Bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices. We show that such factorizations indeed exist over any coefficient ring when the matrix has even size. Establishing a correspondence between such factorizations and extensions of maximal Cohen--Macaulay modules over the generic determinant, we exhibit all factorizations where one of the factors has determinant equal to the generic determinant. The classification shows not only that the Cohen--Macaulay representation theory of the generic determinant is wild in the tame-wild dichotomy, but that it is quite wild: even in rank two, the isomorphism classes cannot be parametrized by a finite-dimensional variety over the coefficients. We further relate the factorization problem to the multiplicative structure of the \Ext--algebra of the two nontrivial rank-one maximal Cohen--Macaulay modules and determine it completely.Comment: 44 pages, final version of the work announced in math.RA/0408425, to appear in the American Journal of Mathematic

Morita Contexts, Idempotents, and Hochschild Cohomology - with Applications to Invariant Rings -

We investigate how to compare Hochschild cohomology of algebras related by a Morita context. Interpreting a Morita context as a ring with distinguished idempotent, the key ingredient for such a comparison is shown to be the grade of the Morita defect, the quotient of the ring modulo the ideal generated by the idempotent. Along the way, we show that the grade of the stable endomorphism ring as a module over the endomorphism ring controls vanishing of higher groups of selfextensions, and explain the relation to various forms of the Generalized Nakayama Conjecture for Noetherian algebras. As applications of our approach we explore to what extent Hochschild cohomology of an invariant ring coincides with the invariants of the Hochschild cohomology.Comment: 28 pages, uses conm-p-l.sty. To appear in Contemporary Mathematics series volume (Conference Proceedings for Summer 2001 Grenoble and Lyon conferences, edited by: L. Avramov, M. Chardin, M. Morales, and C. Polini

The adjoint of an even size matrix factors

We show that the adjoint matrix of a generic square matrix of even size can be factored nontrivially, answering a question of G. Bergman. This note is a preliminary report on work in progress.Comment: 7 pages, preliminary versio